The entanglement of assistance \(E_a\)  and coherence of assistance \(C_a\)  depend on the choice of the functions \(f\in \mathfrak {F}\setminus \lbrace 0\rbrace \) .
If \(f(\vec{p})=-\sum _i p_i \log p_i\)  for \(\vec{p}=(p_1,p_2,\cdots ,p_n)^T\)  in the probability simplex, \(E_a\)  becomes the entanglement of formation of assistance [1]}, [2]}, [3]}, and \(C_a\)  becomes the relative entropy coherence of assistance [4]}.
If \(f(\vec{p})=\sum _{i\ne j}\sqrt{p_ip_j}\)  for \(\vec{p}=(p_1,p_2,\cdots ,p_n)^T\)  in the probability simplex, then \(E_a\)  becomes the half convex-roof
extended negativity of assistance [5]} and \(C_a\)  becomes the \(l_1\)  norm coherence of assistance [6]}. Analogously, one can also define various other types of entanglement of assistance and coherence of assistance based on other real symmetric concave functions \(f\) .
For example, let \(f(\vec{p})=\sqrt{2(1-\sum _i p_i^2)}\) , then \(E_a\)  is the entanglement of assistance in terms of concurrence [7]}, in which an upper bound of entanglement of assistance is provided as \(E_a(\rho )\le \sqrt{2(1-tr(\rho _A^2))}\)  with \(\rho _A=tr_B (\rho )\) . For this function \(f\) , we can define the coherence of assistance \(C_a\)  in terms of concurrence similarly and one upper bound is \(C_a(\rho )\le \sqrt{2(1-\sum _i \rho _{ii}^2)}\)  with \(\rho _{ii}\)  the diagonal entries of \(\rho \)  in the reference basis.
